令 G=(V,E) 為一個包含點集合V及邊集合E的圖,l為將圖G點集合和邊集合對應到一個整數集 {0,…,λ} 的函數使得相鄰的點不能標記相同的整數,相鄰的邊不能標記相同的整數,以及相鄰的點和邊標號差值的絕對值必須大於等於p則稱l為圖G的一個(p,1)-全標號。 在一個(p,1)-全標號中,兩個標記整數之間最大的差值稱為跨度。在圖G的(p,1)-全標號中,最小的跨度我們稱之為圖G的(p,1)-全標號數,以符號λ_p^T (G)表示之。 在這篇論文中,我們證明了對任一Δ-正則圖G以及每一個整數k≥4,如果p≥max{k+1,Δ}且G是Class 1,則χ(G)=k若且為若λ_p^T (G)=Δ+p+k-2.
Let G=(V,E) be a graph. A (p,1)-total labeling of G is a mapping form V(G)∪E(G) into {0,…,λ} for some integer λ such that any adjacent vertices of G are labeled with distinct integers, any two adjacent edges of G are labeled with distinct integers and a vertex and its incident edge receive integers that differ by at least p in absolute value. The span of a (p,1)-total labeling is the maximum difference between two labels. The (p,1)-total number of a graph G is the minimum span of a (p,1)-total labeling of G, denoted by λ_p^T (G). In this thesis, we prove that for each connected Δ-regular graph G and each integer k≥4, if p≥max{k+1,Δ} and G is Class 1, then χ(G)=k if and only if λ_p^T (G)=Δ+p+k-2.
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